What does the term "internet 2" refer to when designing a web page...precisely, how is that different than internet 1 sites?

Um… Do you mean Web 2.0?

Yes…allow me to clarify, I mean web 2.0…sorry I’m not that hip to get the nuance…

Web 2.0 is jargon for websites that are based on user generated content, like facebook, fluther, etc.

Web 2.0 refers to a collection of features that newer web pages had or have since about 2004. The term was probably coined by Tim O’Reilly who used it to describe “new” features in web applications.
Since then the term has been used for different features and is not really a definition on a specific set of features. Very often web pages using AJAX to create a better user experience are described as Web 2.0 (e.g. google mail), but there are quite a few more idicators.

If you want to consider a technologically step from “web 1” to Web 2.0, think of the following: traditionally web pages were either static html pages or used some kind of query/result process e.g. searching for something. This means that when a new page is rendered, the old page disappears and the complete page is loaded and redrawn. I would call this web 1, this has a few drawbacks, most importantly the load time of the page that displays the result. With AJAX or Javascript DOM manipulation, you can change parts of the page based on user input, search results or other state changes without reloading the page, I would call this Web 2.0. (For example the Live Preview feature of Fluther is very much a Web 2.0 feature since it modifies the DOM of the page instantly)

In a broader sense, Web 2.0 is used for features including user contributed content, online video, tagged information, tag clouds, whats related lists, rss syndication etc.